[Proof of Theorem  and ]
Let \(H_R = \left\lbrace  h_{i_1}, \ldots , h_{i_{k_n}} \right\rbrace \)  be the set of selected nodes by DPP Node pruning method. Recall from [1]} that every student hidden node specializes in learning a teacher node. Denote \(t(h)\)  to be the teacher node learnt by \(h\) . \(S_{m}\subseteq H_R\)  be the set of selected hidden nodes of the pruned network which learnt the \(m^{th}\)  teacher node , i.e., \(S_m = \lbrace h \in H_R | t(h) = t_m \rbrace \)  (\(t_m\)  is the \(m^{th}\)  teacher node). Hence, \(prn = |\lbrace {1}(|S_m|>0)| 1 \le m \le M \rbrace | \)  is the number of teacher nodes explained by the pruned network and W.L.O.G. we can assume that \(t_1, \ldots , t_{prn}\)  are those set of teacher nodes. Let \(l_1, \ldots , l_{prn}\)  be the number of student nodes in the pruned network which learn the corresponding teacher node. Note that, \(\sum _{i=1}^{prn} l_i = k_n\)  and \(l_i\le Z\)  (where \(Z\)  is the number of student nodes dedicated to learn a single teacher node in the unpruned network) for all \(i\) . Applying Lemma REF  directly we can see that the GE for the pruned network is
\(\frac{(v^*)^2}{6}\left[ \sum _{i=1}^{prn}\left(1-\frac{l_i}{Z} \right)^2\right] + \frac{(M-prn)(v^*)^2}{6}\) 
